科学与数学eev.ee
写给极客的音乐理论
Music theory for nerds
2016 — 2022，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
这篇博客以程序员和数学爱好者能理解的方式介绍乐理，从音高、音程和音阶的数学关系讲起，试图解释西方记谱与调性体系，并讨论为什么是十二个音等问题。评论区则围绕大小调是否等同、记谱法的意义以及纯律到十二平均律的过渡展开补充和争论。
为什么经久不衰：它把乐理从死记硬背变成可推导的模型，而高赞评论又补上调性、记谱和律制等关键细节，因此多年后仍值得一读。
HN 评论精选
音阶不只是音符集合，主音/起始点决定调性，大小调不能等同。
我想以同样“给书呆子”的方式补充一下 Eevee 似乎在音阶和记谱上有点迷失的地方。他（她？不确定）把 A 小调和 C 大调音阶称为相同，因为它们包含相同音符。这想法不奇怪，但就像因为正弦和余弦包含相同的一组值、顺序也相同，就说它们相同。区别在相位。基本上，音阶不只是有序音符集合，还有起点。这个音，即音阶得名的音，常是该调曲子最后一个音，也常是第一个音。所以如果你按 Eevee 建议用 C 大调弹《致爱丽丝》，整个旋律会比用 A 大调弹高三个半音，而且听起来别扭，因为本应是小调。理解这点后，整个记谱体系也合理多了。
skrebbel查看英文原文
I'd like to elaborate a bit, in the same "for nerds" manner, on where Eevee seems to get lost a bit with scales and notation. He (she? not sure) calls the A minor and C major scales the same, because they contain the same notes. That's not an odd thought, but it's like calling sine and cosine the same because both functions contain the same set of values, in the same order. The difference is phase. Basically, scales aren't just an ordered set of notes, they also have a starting point. This note, the note the scale is named after, is often the last note of a tune played in that scale, and often the first too (especially for more poppy tunes). So if you play Für Elise in C major, like Eevee suggests, the entire melody will be pitched 3 semitones higher than playing Für Elise in A major. And it'll sound awkward because you're supposed to play it on a minor scale. Once you understand this, the whole notation thing makes a lot more sense as well.
记谱法通过音符位置帮助识别和弦和音程，调性由终止和弦决定。
“完全掩盖了音高关系”其实没有掩盖，反而更清楚。比如看到 C、E、G 和临时记号，我知道应弹 C 三和弦；不同三和弦类型靠上下文选。若应弹 C# 大三和弦，写成 C#-F-G# 而不是 C#-E#-G# 会令人困惑，看起来像 sus4 琶音。所以五线谱上音阶的写法促进对音程关系的模式匹配。C 大调和 A 小调不相同：音阶音符相同，但曲子结束在 C 大三和弦说 C 大调，结束在 A 小三和弦说 A 小调，这对理解演奏很重要。
theseoafs查看英文原文
> It completely obscures the relationship between the pitches, though. It doesn't actually obscure the relationship between the notes -- it makes them clearer. For example, I see the notes C, E, and G on some sheet music, maybe with some accidentals on some of those notes. I know that I'm therefore supposed to play a C triad. Now, there are multiple kinds of triads, but once I know I'm supposed to play a triad, it's easy to use context to pick out which one I need (major, minor, diminished, augmented -- usually one of the first two). If I were supposed to play a C# major triad, though, and the written notes were (C#-F-G#) as opposed to what they should be (C#-E#-G#) then that's confusing because it looks like I should be playing an arpeggiated sus4 of some kind. So the written nature of scales on the staff engenders an understanding of the relationship between the notes. Basically we write things the way we do so that the people reading the music can more efficiently pattern-match. > C major is identical to A minor, and I don’t understand why we need both. They're not identical. C major and A minor have the same notes in their respective scales. But we say that a piece is in the key of C major when it resolves to the a C major chord at the end, and we say a piece is in the key of A minor when it resolves to an A minor chord at the end -- an important concept for reasoning about how a piece is supposed to be performed. > C minor: C D D# F G G# A# C Eb, Ab, and Bb, not D#, G#, and A#. > This has got to be some of the worst jargon and notation for anything, ever. It's really not. Keep practicing. It makes sense, I promise. I hear a lot of people -- usually people who have not been studying music for very long -- insist that the system would be more logical if there were no accidentals and there were 12 notes with distinct names and the staff had a bunch more lines on it. I've never bought it. The notation of music isn't arbitrary, it's informed by experience and it works.
用相对音级/唱名理解音乐比记绝对音名更有效。
许多比我强得多的音乐家惊讶我能只听电台就弹歌。突破在于理解音符真正的“意义”在于相对主音的位置（I-II-III，do-re-mi）。这样杂乱消失，问题可控。三音曲“do re mi”在 C 调是 C-D-E，在 G 调是 G-A-B，同一旋律每个频率同比例增加。试图靠字母理解音乐，像读每篇文章都被换成不同“密钥”加密的世界：正常字母表里“cab”在“B 调”写成“dbc”，只有意识到 d 代表第四个字母才能解出意义；d 本身无意义，而“字母表第 4 个”有意义。把音乐破译成 I、II、III、IV、V，复杂度就 manageable，可学习辨认 III 音或 VI 小和弦声音。大调只有八个音。
karlb查看英文原文
Many musicians much better than me are surprised at how I can play a song just by hearing it on the radio. My breakthrough came from understanding music was realizing that the real “meaning” of a note lies in its position relative to the tonic note (e,g, I-II-II, etc, also written do-re-mi). Suddenly, almost all of the clutter was removed, and the problem became manageable. Let's consider the three-note tune “do, re, mi”. If that tune were played in the key of C, it would become C-D-E. If it were played in G, it would become G-A-B. But in either case, it's the same tune but with each frequency increased by the same percentage. Trying to understand music by understanding the letters is like trying to read in a world where every article has been enciphered into a different “key”: e.g., the word "cab" in “the key of A” (the alphabet we normally use) would be written as "dbc" if the article were written in “the key of B”. In the latter case, you could discern meaning only once you realised that the letter “d” represented the third letter of the alphabet. There's nothing meaningful about a “d” but there is something meaningful about a “4th letter of the alphabet”. Once you start to “decipher” all music into I, II, III, IV, V, etc., the complexity becomes manageable. You can start to learn to recognize the sound of a III note, or of a VI minor chord. After all, there are only eight notes in the major scale.
文章缺少从纯律到十二平均律的过渡解释，12 音是折中方案。
关于“为什么是 12 个音”，强烈推荐 David Bennett 的视频。文章缺少一个关键步骤：从谐波系列定义的纯律音符（基频的整数倍比率，自然物理现象）到大多数西方音乐使用的十二平均律，后者调整比率以便使用某些作曲技巧而不显刺耳。纯律从不同起始音定义音阶会得到不同频率，比如从 A 和从 B 开始得到的“下一个 E”不同。
PaulDavisThe1st查看英文原文
On the question of "why 12 notes", I STRONGLY recommend this video by David Bennett: https://www.youtube.com/watch?v=lvmzgVtZtUQ There's a really important step missing in the article which has been aluded to by a couple of comments already. It's the step between: 1. a series of notes defined by the harmonic series, that is, exact integer multiples (or ratios, if you prefer) of a base frequency, a physical phenomena found in nature and 2. the well-tempered scales used by most western music, which adjusts the ratios to allow certain compositional "tricks" to be used without sounding dissonant. The set of notes defined by (1) are typically referred to as "Just Intonation", and was the basis for most western music (and some non-western music) until somewhere between about 1400 and 1600 (lots of room for discussion/debate there). The "problem" with just intonation, if indeed it is a problem, is that if you define two series of notes (call them a scale, if you like) using these integer ratios but starting from two different notes, you will trivially find cases where "the next higher E" is a different frequency. So starting from (say) A and moving by integer ratios gets you to a different (say) F than if you start from (say) B. This means that if you are writing in a just intonation scale with (say) A as the root, you have a set of notes that are not actually the same frequencies as if your scale started on (say) B. By itself, this is not a problem at all - there is all kinds of lovely music written through history that works just fine with notes and scales defined this way. You just stay in the same scale throughout, and there are no issues. There are even some scale changes you can make that still work, you just have to know what they are (and they depend on the root note and the set of integer ratios you're using, so it gets complicated). But ... somewhere in the period mentioned above, a subset of western musical culture started to want to experiment with "modulation" - changing from one scale to another in the midst of piece. On continuous pitch instruments (e.g. violin, voice), this is entirely possible to do, since they can play any frequency in their range at all. However, it does require significant skill on the part of the performer, since the pitch of a (say) "F" will differ depending on the scale currently in use in the piece. The breaking point, such as it was, came with the development of fixed pitch instruments (keyboards). These can only play the notes they are currently tuned to, and so if the (say) F in two scales is a different frequency, you cannot play in both scales without retuning - an obvious impossibility in the middle of a piece. So, "well-tempered" tuning was developed - the ratios described in TFA. These are tweaked by relatively small amounts so that the notes are close to where Just Intonation (integer ratios) would have placed them, but not precisely the same. These small shifts mean that the (say) "F" is the same frequency whether your scale began on (say) A or B. You can now play a fixed-tuning instrument like a harpsichord or a piano, change from one scale to another, and everything remains "in tune". Of course, to ears used to Just Intonation, "well-tempered" tuning sounds out of tune. But in the west, most people (even within our musical academies) have grown up used to the sound of well-tempered tuning, and it is Just Intonation that sounds "off".
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